# Definition(s) > [!NOTE] Definition 1 (Separately Continuous Real-Valued Function of Two Variables) > Given a real-valued function $f:U \subset \mathbb{R}^2 \to \mathbb{R}$, we consider two families of functions $\{ g^y : \mathbb{R}\to \mathbb{R} \}_{y\in \mathbb{R}}$ and $\{ h^x: \mathbb{R} \to \mathbb{R} \}_{x\in \mathbb{R}}$ defined by $g^y(x) := f(x,y) =: h^x(y).$ > We say that $f$ is separately continuous at $(x_{0},y_{0})$ if $g^{y_{0}}(x)$ is [[Continuous Real Function|continuous]] at $x_{0}$ as a function of $x$ and $h^{x_{0}}(y)$ is continuous at $y_{0}$ as a function of $y$. > [!Example] Example > Contents # Properties(s) # Application(s) **More examples**: # Bibliography